A numerical study of the rheological properties of suspensions of ...

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180 M. B. Mackaplow and E. S. G. ShaclfehVolume fraction of fibres, p0.01 0.02 0.03 0.04 0.05I I I I In13FIGURE 10. Steady-state shear viscosity of a suspension of cylindrical fibres with aspect ratio,A = 33, relative to that of the fibre-free liquid, as a function of suspension volume fraction. Shownare the the experimentally determined values of Bibbo (1987), and the predictions of numericalsimulations, the dilute theory, and the semi-dilute theory of Shaqfeh & Fredrickson (1990).similar orientation, and at steady state most fibres in shear are aligned in the flowdirection, such a modification to our algorithm is necessary to prevent biasing ourdistribution.In figure 10 we have compared our simulated results for the steady-state shearviscosity for suspensions of particles having aspect ratios of 33 to the predictions ofthe dilute theory and the semi-dilute theory of Shaqfeh & Fredrickson (1990). Boththeories make use ( ~ 1 ~ 1 ~ = 2 ~ 0.100, 2 ) as determined from our generated orientationdistributions. For the semi-dilute theory, since the O(1) constant has not beendetermined for this particular orientation distribution, we set it equal to zero.In figure 10 we see fairly good agreement between simulations and experiments up toconcentrations of approximately n13 = 7, above which the simulations underestimatethe experimentally determined shear viscosity. Based on the previously discussedresults for isotropic suspensions, and the fact that by symmetry ( ~ 1 ~ 1 ~ = 2 ~ 0, 3 we )believe that the curved streamlines, no-slip condition on the outer wall, and shear inthe (2, 3)-plane, all present in the experimental apparatus, had a negligible effect onthe shear viscosity and would not account for the discrepancy between simulationand experiments.The likely sources of the discrepancy can be divided into two classes: thosewhich are equally important at all suspension concentrations and those which aremore important at higher concentrations. Concerning the former, there are threelikely factors. First, a fibre oriented in the flow direction will make a contributionto the suspension stress tensor 0 (lnA/A2) smaller than one which is not. Sincethe contribution to the stress of the former is driven entirely by gradients in theundisturbed velocity along the cross-sections of the fibres, it will not be captured bythe slender-body theory approximation used in the simulations. For fibres in shearflow at steady state, only an 0(1/A) fraction of the fibres are not approximatelyaligned in the flow direction (Stover et al. 1992). Thus, the error induced by theslender-body theory approximation in calculating the steady-state shear viscosity is
Rheological properties of suspensions of rigid, non-Brownian fibres 181O(ln A/A). This is not necessarily small for the suspensions under consideration, since(lnA)/A fi: 0.11. Secondly, when comparing suspensions of the same aspect ratio,(p1plp2p~) for our numerically generated suspensions was about 10% smaller than theexperimentally determined values of Stover et al. (1992), due to small changes in theJeffrey orbits induced by fibre interactions. Finally, (p1p1p2p2) in the experiments ofBibbo (1987) were probably even larger than those measured by Stover et al. (1992),since the effect of the aforementioned shear in the (2, 3)-plane would be to rotatefibres into the (1, 2)-plane.It is possible to suggest a reason for the relatively poorer agreement betweensimulations and experiments for n13 > 5. We expect the average closest approachdistance in the suspension will be between those for aligned and isotropic suspensions.However, from an analysis similar to that used by Doi & Edwards (1989), we expectthe O(l/A) fraction of the fibres that are not aligned in the flow direction willexperience a closest approach distance of the same order as that in an isotropicsuspension. Since it is these few fibres that make the dominant contribution to(c!;’), and we see a decline in the agreement between simulations and experimentsat approximately the same concentration as for isotropic suspensions, we suggest thatthe discrepancy is due to slender-body theory underestimating the effects of closefibre-fibre interactions.6. ConclusionFor suspensions of rigid, non-Brownian fibres at zero Reynolds number, hydrodynamicfibre-fibre interactions have a negligible effect on the volume-averaged stresstensor for n13 < 1. At n13 - O(l), independent of any particular fibre aspect ratio orsuspension volume fraction, fibre-fibre interactions begin to enhance the stress in thesuspension and it undergoes a transition to the semi-dilute regime. This transition,as well as the suspension behaviour well into the semi-dilute regime, is well predictedby dilute theories that take into account two-body interactions. In the semi-diluteregime, the dimensionless fibre disturbance screening length, x/b, is only a functionof suspension volume fraction. It is approximately the same for both aligned andisotropic suspensions, even though the latter contain many more close fibre-fibreinteractions than the former. This is in qualitative agreement with the theoreticalprediction of Shaqfeh & Fredrickson (1990), but disagrees with that of Dinh & Armstrong(1984). Our conclusion is supported by the experimental work of Bibbo (1987),who measured the zero-shear viscosity of an isotropic suspension. Bibbo (1987) hadhypothesized that rapid alignment of the suspension caused an initially isotropicsuspension to show the same screening behaviour as that expected for an alignedsuspension, even at very short times. Our numerical simulations have shown thatpartial alignment of the suspension is not necessary for this screening behaviour tooccur. The quantitative agreement between our simulations and the semi-dilute theoryof Shaqfeh & Fredrickson (1990) can be greatly improved by adjusting the value ofC” in the theory to -1.25 and retaining the O(l/ln(l/+)) term with a coefficient of2.4. This physically corresponds to keeping classes of fibre interactions neglected bythe theory. The above result is used to show that the suspension screening lengths arevery similar to those in fibrous porous media having the same fibre volume fraction.This holds for both aligned and isotropic orientation distributions. We also directlyverify suspension screening by studying the decay of the velocity fields created byindividual Stokeslets in fibre suspensions.
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